source: ncert
topic: probability
class_level: class_12
chapter: ch13
difficulty: advanced
Class 12 | Ch13: Probability CONDITIONAL PROBABILITY: P(A|B)=P(A∩B)/P(B) where P(B)≠0. P(A∩B)=P(A|B)P(B)=P(B|A)P(A). MULTIPLICATION THEOREM: P(A∩B)=P(A)P(B|A). P(A∩B∩C)=P(A)P(B|A)P(C|AB). INDEPENDENT EVENTS: A and B independent: P(A∩B)=P(A)P(B). P(A|B)=P(A). A,B,C independent: P(A∩B)=P(A)P(B), P(B∩C)=P(B)P(C), P(A∩C)=P(A)P(C), AND P(A∩B∩C)=P(A)P(B)P(C). (All four conditions needed.) TOTAL PROBABILITY THEOREM: If B1,B2,...,Bn are mutually exclusive and exhaustive events: P(A)=P(A|B1)P(B1)+P(A|B2)P(B2)+...+P(A|Bn)P(Bn). BAYES' THEOREM: P(Bi|A)=P(A|Bi)P(Bi) / [P(A|B1)P(B1)+...+P(A|Bn)P(Bn)]. Prior probability: P(Bi). Posterior probability: P(Bi|A). RANDOM VARIABLE AND PROBABILITY DISTRIBUTION: Random variable X: function from sample space to real numbers. Mean (Expected value): E(X)=mu=sum(xiP(xi)). Variance: Var(X)=E(X^2)-[E(X)]^2=sum(xi^2P(xi))-mu^2. SD=sqrt(Var(X)). BERNOULLI TRIALS AND BINOMIAL DISTRIBUTION: n independent trials, each with probability p of success. P(X=r)=nCrp^r(1-p)^(n-r)=nCrp^rq^(n-r) where q=1-p. Mean=np. Variance=npq. SD=sqrt(npq). SOLVED EXAMPLES: Example 1: Bag A: 3R,5B. Bag B: 4R,6B. One bag chosen randomly, red ball drawn. Find P(bag A). P(A)=P(B)=1/2. P(R|A)=3/8. P(R|B)=4/10=2/5. P(R)=P(R|A)P(A)+P(R|B)P(B)=(3/8)(1/2)+(2/5)(1/2)=3/16+1/5=31/80. P(A|R)=P(R|A)P(A)/P(R)=(3/16)/(31/80)=(3/16)(80/31)=15/31. Example 2: Binomial X~B(10,0.4). Find P(X=3). P(X=3)=10C3(0.4)^3(0.6)^7=1200.0640.0279936≈0.2150. Mean=100.4=4. Variance=100.40.6=2.4. COMMON MISTAKES: Bayes theorem: denominator is TOTAL probability P(A) using all branches. Binomial: trials must be INDEPENDENT with CONSTANT probability p. E(X^2) ≠ [E(X)]^2. Variance=E(X^2)-[E(X)]^2.